Lily was shown to be rather good at potion-making, as Horace Slughorn comments on several times:
“Hmpf. Yes, well. You shouldn’t have favorites as a teacher, of course, but she was one of mine. Your mother,” Slughorn added, in answer to Harry’s questioning look. “Lily Evans. One of the brightest I ever taught. Vivacious, you know. Charming girl. I used to tell her she ought to have been in my House. Very cheeky answers I used to get back too.”
Harry Potter and the Half-Blood Prince, chapter 4: "Horace Slughorn"
“The clear winner!” he cried to the dungeon. “Excellent, excellent, Harry! Good lord, it’s clear you’ve inherited your mother’s talent. She was a dab hand at Potions, Lily was! Here you are, then, here you are — one bottle of Felix Felicis, as promised, and use it well!”
Harry Potter and the Half-Blood Prince, chapter 9: "The Half-Blood Prince"
So evidently she had a special talent for potion-making. While it's not clear whether this requires magical talent or if it's just as if Lily's a whiz at cooking, it is evidence of an above-average ability in a magical field.
As for James, we have proof that he was rather proficient with magic as well. He accomplished a rather major task at a very young age: becoming an Animagus.
"And they didn’t desert me at all. Instead, they did something for me that would make my transformations not only bearable, but the best times of my life. They became Animagi."
"My dad too?" said Harry, astounded.
"Yes, indeed," said Lupin.
Harry Potter and the Prisoner of Azkaban, chapter 18: "Moony, Wormtail, Padfoot, and Prongs"
Becoming an Animagus is very difficult - at the time that Harry and co. were in school, there were only seven registered Animagi in existence (of course, this was excluding Pettigrew and Sirius).
In any case, this shows that James was indeed capable of quite advanced magic that would be far beyond the reach of many other wizards.
Now that I've shown that both Lily and James had above-average talents in their own ways, I'd also like to point out that this doesn't necessarily contribute to Harry being talented as well. The existence of Squibs proves that having magical parents or powerful parents does not necessarily mean that the children will be powerful - heck, Lily was Muggle-born!
So - yes, both Lily and James had above-average talents, but no, that doesn't necessarily explain why Harry was talented.